224 ANALYTICAL MECHANICS
ILLUSTRATIVE EXAMPLES ON THE MOTION OF A RIGID BODY ABOUT A FIXED AXIS..
Discuss the motion of a rigid body, which is free to rotate about a fixed axis, under the action of a constant torque.
Suppose the body to be the flywheel of Fig. 110. Let the constant. torque be supplied by a constant force F applied at the free end of the string which is wound around the axle. The tensile force of the string and the reaction of the bearings form a couple, the torque of which equals the moment of the force F about the axis of rotation. Therefore
0 = Fa,
where a is the radius of the axle. Substituting this value of G in the torque equation we obtain
du Fa ,
or — = -7- = V = const.
at I
Integrating the last equation we get
co = yt + c. Let a? = co0 when t — 0, then c = co0. Therefore
co = co0 + &t, (1)
or — = o)0 + cot.
Integrating again
B
Let 6 = 0 when t = 0, then c' = 0. Therefore
0 = 0,^ + 1 co£2. (2)
Eliminating t between equations (1) and (2)
0,2 = Wo2 + 2 ye* (3)
ENERGY METHOD. — The increase in the kinetic energy due to the action of F is
r-ro = Uco2-Hco02.
The diminution in the potential energy of the system which supplies the force F equals the work done by F. Therefore
* Compare equations (1), (2), and (3) with the corresponding equations of p. 113.